It is difficult to say whether or not you have a genuine black pearl without personally looking at the strand, but I can give you a few things to look for to determine its origin.
First, what is the size of the pearls in the necklace? If the pearls are less than 9mm it is likely that the strand is not Tahitian. If the strand is less than 8mm, it is 100% not Tahitian.
You have stated that you can see the white core. This is possible even with Tahitian pearls. The nuclei used in Tahitian pearls is like that used in other pearls and is white. But the nacre usually quite thick on a Tahitian strand and it is rare to see the nucleus. If the nucleus is visible, it is a very low quality pearl.
The pearls may be Akoya as well. With thin nacre it is possible to see the nucleus of an Akoya pearl. If you can see an off-white coloration the pearls are dyed. If the nucleus is black, the pearls have been treated with radiation.
The only genuine black pearls are Tahitian. If the pearls are not genuine Tahitian pearls, they are not naturally black. Akoya black pearls and freshwater black pearls are always treated to attain the black coloration. This is absolute, without exception. Keep this in mind when shopping for black pearls. If they tell you the pearls are naturally black, but not Tahitian, you know you are either dealing with a dishonest pearl dealer, or an uneducated pearl dealer.